A common problem in statistics is determining whether or not the means of 2 populations are equal. The independent 2-sample t-test is a popular parametric method to answer this question. (In an earlier Statistics Lesson of the Day, I discussed how data collected from a completely randomized design with 1 binary factor can be analyzed by an independent 2-sample t-test. I also discussed its possible use in the discovery of argon.) I have learned 2 versions of the independent 2-sample t-test, and they differ on the variances of the 2 samples. The 2 possibilities are

equal variances

unequal variances

Most statistics textbooks that I have read elaborate at length about the independent 2-sample t-test with equal variances (also called Student’s t-test). However, the assumption of equal variances needs to be checked using the chi-squared test before proceeding with the Student’s t-test, yet this check does not seem to be universally done in practice. Furthermore, conducting one test based on the results of another can inflate the probability of committing a Type 1 error (Ruxton, 2006).

Some books give due attention to the independent 2-sample t-test with unequal variances (also called Welch’s t-test), but some barely mention its value, and others do not even mention it at all. I find this to be puzzling, because the assumption of equal variances is often violated in practice, and Welch’s t-test provides an easy solution to this problem. There is a seemingly intimidating but straightforward calculation to approximate the number of degrees of freedom for Welch’s t-test, and this calculation is automatically incorporated in most software, including R and SAS. Finally, Welch’s t-test removes the need to check for equal variances, and it is almost as powerful as Student’s t-test when the variances are equal (Ruxton, 2006).

For all of these reasons, I recommend Welch’s t-test when using the parametric approach to comparing the means of 2 populations.

I learned about Lord Rayleigh’s discovery of argon in my 2nd-year analytical chemistry class while reading “Quantitative Chemical Analysis” by Daniel Harris. (William Ramsay was also responsible for this discovery.) This is one of my favourite stories in chemistry; it illustrates how diligence in measurement can lead to an elegant and surprising discovery. I find no evidence that Rayleigh and Ramsay used statistics to confirm their findings; their paper was published 13 years before Gosset published about the t-test. Thus, I will use a 2-sample t-test in R to confirm their result.